The Gauss-Conform map projection is literally in continuous daily use in many of the programs used by Remote sensing and GIS people. I have found that many of the problems I have had to solve over the last few years have related to these programs. But I found that the knowledge I had and could glean from the available textbooks, manuals, papers etc., was insufficient to actually understand what the Gauss-Conform projection is.
Most of the available literature on the subject I have found reminded me of the cartoon of a blackboard full of complicated equations. One of the steps in the derivation was, ``And then a miracle happens!'' Some of the literature is just plain wrong. Some of it is in German. Some of it is scattered across several volumes of an ancient survey journal mixed with accounts of the hazards the early surveyors encountered. Fascinating reading, but the mathematics is faintly quaint and archaic.
It is notable how much ingenuity was invested in those times in making it simpler for field surveyors to perform the necessary calculations. In the days before computers this sophistication was a blessing, now the only effect of that same ingenuity is to make things more complicated and to get in the way of understanding.
Some comprehensive textbooks exist, but I found them almost as painful, as they set out to teach you the whole field of geodesy with the Gauss Conform Projection as a special case of the general theory. This approach requires one to read the entire book to glean the small bit you are really interested in. Even so, one is left with doubts as to whether the book is actually treating the projection used in South Africa.
Whilst the modern image processing systems and GIS's come equipped with a vast battery of preprogrammed numerical routines, I would strongly advocate that the user also be equipped with a good algebraic manipulation package. Just as calculators and spreadsheets and packages have vastly improved our numerical calculation abilities, an algebraic manipulation package can create equally large quantum leaps in our ability to tackle complex problems symbolically. Although I stated that this is to be a worked example easy to follow, actually it is a lot easier to follow if you have an algebraic manipulation package! Especially the Taylor expansions and the derivatives become a lot easier.